freealg.AlgebraicForm.stieltjes

AlgebraicForm.stieltjes(x=None, y=None, plot=False, latex=False, save=False)

Compute Stieltjes transform of the spectral density

Parameters:

xnumpy.array, default=None

The x axis of the grid where the Stieltjes transform is evaluated. If None, an interval slightly larger than the support interval of the spectral density is used.

ynumpy.array, default=None

The y axis of the grid where the Stieltjes transform is evaluated. If None, a grid on the interval [-1, 1] is used.

plotbool, default=False

If True, density is plotted.

latexbool, default=False

If True, the plot is rendered using LaTeX. This option is relevant only if plot=True.

savebool, default=False

If not False, the plot is saved. If a string is given, it is assumed to the save filename (with the file extension). This option is relevant only if plot=True.

Returns:

mnumpy.ndarray

The Stieltjes transform on the principal branch.

See also

density

hilbert

plot_branches

Notes

The Stieltjes transform is defined by

\[m(z) = \int_{\mathbf{R}} \frac{\rho(x)}{x - z} \, \mathrm{d}x.\]

The Stieltjes transform is a Herglotz function, meaning

\[\Im(m(z)) > 0, \quad z \in \mathbb{C}^{+}.\]

Also, it satisfies normalization at infinity:

\[m(z) = -\frac{1}{z}, \qquad \vert z \vert \to \infty.\]

This function evaluates Stieltjes transform on an array of points, or over a 2D Cartesian grid on the complex plane.

Examples

>>> # Create a distribution with two bulks
>>> from freealg.distributions import CompoundFreePoisson
>>> cfp = CompoundFreePoisson(t=[2.0,  5.5], w=[0.75, 0.25],
...                           lam=0.1)

>>> # Create AlgebraicForm and fit the distribution
>>> from freealg import AlgebraicForm
>>> af = AlgebraicForm(cfp)
>>> af.fit(deg_m=3, deg_z=1)

>>> # Plot the density of the fitted spectral curve
>>> import numpy
>>> x = numpy.linspace(0, 8, 100)
>>> y = numpy.linspace(-2, 2, 100)
>>> m = af.stieltjes(x, y, plot=True)